Rings of functions on non-abelian groups

For several classes of finite nonabelian groups we investigate the structure of the ring of functions, $\mathcal R(C)$, determined by the cover $C$ of maximal abelian subgroups. We determine the Jacobson radical $J(\mathcal R(C))$ and the semisimple quotient ring $\mathcal R(C)/J(\mathcal R(C))$.